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Fundamentals of Digital Signal Processing

DFT Linearity

Learn how the linearity property of the Discrete Fourier Transform (DFT) allows the DFT of a sum of signals to be expressed as the sum of their individual DFTs. This lesson covers the derivation and a practical example demonstrating how complex signals can be analyzed efficiently using this property.

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Linearity implies that the DFT of the sum of input signals is the sum of the DFTs of the individual input signals. Let’s find out how.

Derivation

Assume that two input signals x1[n]x_1[n] and x2[n]x_2[n] have DFTs given by X1[k]X_1[k] and X2[k]X_2[k], respectively. If that’s the case, the DFT of their sum is equal to the sum of their individual DFTs.

DFT x1[n]→X1[k]DFT x2[n]→X2[k]\begin{align*} \text{DFT }x_1[n]\quad \rightarrow\quad X_1[k]\\ \text{DFT }x_2[n]\quad \rightarrow\quad X_2[k] \end{align*} ...